Subgroups — #16854628

Order 3968.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521816	C₂	1	2	yes	maximal, complemented
2	#14521171	C₂	1	4	yes	maximal, complemented
3	#14521166	C₂	1	4	yes	maximal, complemented
4	C₂×#11758407	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521066	C₂	1	2	yes	maximal, complemented
6	#14521004	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
8	C₂×#41×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#135	C₄	1	2	yes	complemented
10	C₃₁×#135	C₄	1	2	yes	complemented
11	C₃₁×#135	C₂²	1	2	yes	—
12	C₃₁×#135	C₂²	1	2	yes	—
13	C₂×Q₁₆×C₃₁	C₄	1	1	yes	characteristic, complemented
14	C₂×SD₁₆×C₃₁	C₄	1	2	yes	complemented
15	C₂×D₈×C₃₁	C₄	1	1	yes	characteristic, complemented
16	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	#11758525	—	2	4	no	—
18	#11758525	C₂²	1	8	yes	complemented
19	#11758503	C₂²	1	1	yes	characteristic
20	C₂×#92474	—	2	4	no	—
21	C₂×#92474	C₂²	1	2	yes	—
22	C₂×#92474	C₂²	1	1	yes	characteristic
23	#11758425	C₂²	1	8	yes	complemented
24	#11758407	C₂²	1	2	yes	complemented
25	#11758407	C₂²	1	2	yes	complemented
26	#11758403	C₂²	1	2	yes	complemented
27	C₈×Dic₃₁	C₂²	1	2	yes	complemented
28	#41×C₃₁	—	2	2	no	—
29	#41×C₃₁	—	2	2	no	—
30	#41×C₃₁	C₂×C₄	1	4	yes	—
31	#41×C₃₁	C₂³	1	4	yes	—
32	C₂×Q₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×D₄×C₃₁	—	2	2	no	—
34	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₃₁	—	2	2	no	—
36	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
38	Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
39	SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
40	SD₁₆×C₃₁	C₂×C₄	1	4	yes	—
41	D₈×C₃₁	C₂×C₄	1	2	yes	—
42	D₈×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²⋊Dic₃₁	—	2	1	no	—
50	C₄⋊Dic₃₁	—	2	1	no	—
51	C₄×Dic₃₁	C₂³	1	2	yes	—
52	#92474	—	2	2	no	—
53	#92474	—	2	8	no	—
54	#92474	D₄	1	2	yes	—
55	#92474	D₄	1	2	yes	—
56	#92474	C₂³	1	4	yes	—
57	C₂×C₃₁⋊C₈	—	2	2	no	—
58	C₂×C₃₁⋊C₈	—	2	8	no	—
59	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
60	C₂³×C₃₁	—	2	2	no	—
61	C₂³×C₃₁	#41	1	1	yes	characteristic
62	Q₈×C₃₁	—	2	2	no	—
63	Q₈×C₃₁	C₂²×C₄	1	4	yes	—
64	D₄×C₃₁	—	2	4	no	—
65	D₄×C₃₁	—	2	4	no	—
66	D₄×C₃₁	—	2	2	no	—
67	D₄×C₃₁	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	2	4	no	—
70	C₂×C₄×C₃₁	—	2	4	no	—
71	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₃₁	—	2	2	no	—
78	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₃₁⋊C₈	—	2	4	no	—
83	C₃₁⋊C₈	C₂×D₄	1	4	yes	—
84	#2957	—	31	1	no	maximal, Sylow
85	C₂²×C₃₁	—	2	4	no	—
86	C₂²×C₃₁	—	4	1	no	—
87	C₂²×C₃₁	—	8	2	no	—
88	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
89	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
90	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
91	C₄×C₃₁	—	2	4	no	—
92	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
93	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
94	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	Dic₃₁	—	4	2	no	—
97	C₂×#135	D₃₁	1	1	yes	characteristic
98	#568	—	31	2	no	—
99	#444	—	31	4	no	—
100	#443	—	31	4	no	—
101	C₂×#108	—	31	1	no	—
102	#421	—	31	2	no	—
103	#405	—	31	1	no	—
104	C₂×C₃₁	—	2	1	no	—
105	C₂×C₃₁	—	2	1	no	—
106	C₂×C₃₁	—	2	1	no	—
107	C₂×C₃₁	—	4	4	no	—
108	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
109	C₂×#41	C₂×D₃₁	1	2	yes	—
110	#135	Dic₃₁	1	2	yes	complemented
111	#135	Dic₃₁	1	2	yes	complemented
112	#135	C₂×D₃₁	1	2	yes	—
113	#135	C₂×D₃₁	1	2	yes	—
114	C₂×Q₁₆	Dic₃₁	1	1	yes	characteristic, complemented
115	C₂×SD₁₆	Dic₃₁	1	2	yes	complemented
116	C₂×D₈	Dic₃₁	1	1	yes	characteristic, complemented
117	#131	—	31	8	no	—
118	#131	—	62	4	no	—
119	C₂×M₁₆	—	31	1	no	—
120	C₂×M₁₆	—	31	2	no	—
121	C₂×M₁₆	—	62	4	no	—
122	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
123	#117	—	31	1	no	—
124	#108	—	31	2	no	—
125	#108	—	31	2	no	—
126	#104	—	31	8	no	—
127	#97	—	31	2	no	—
128	C₄×C₈	—	31	2	no	—
129	C₃₁	#2957	1	1	yes	characteristic, complemented, Sylow
130	#41	—	2	2	no	—
131	#41	—	2	2	no	—
132	#41	C₂×Dic₃₁	1	4	yes	—
133	#41	C₂²×D₃₁	1	4	yes	—
134	C₂×Q₈	C₂×Dic₃₁	1	2	yes	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	C₂×Dic₃₁	1	2	yes	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
139	Q₁₆	C₂×Dic₃₁	1	2	yes	—
140	Q₁₆	C₂×Dic₃₁	1	2	yes	—
141	SD₁₆	C₂×Dic₃₁	1	4	yes	—
142	SD₁₆	C₂×Dic₃₁	1	4	yes	—
143	D₈	C₂×Dic₃₁	1	2	yes	—
144	D₈	C₂×Dic₃₁	1	2	yes	—
145	M₁₆	—	31	2	no	—
146	M₁₆	—	31	2	no	—
147	M₁₆	—	31	4	no	—
148	M₁₆	—	62	8	no	—
149	M₁₆	—	62	2	no	—
150	C₂×C₈	—	31	2	no	—
151	C₂×C₈	—	62	8	no	—
152	C₂×C₈	—	62	2	no	—
153	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
154	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
156	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
159	C₄⋊C₄	—	62	1	no	—
160	C₂²⋊C₄	—	62	1	no	—
161	C₄²	—	31	2	no	—
162	C₂³	—	2	2	no	—
163	C₂³	#92497	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	Q₈	C₂²×Dic₃₁	1	4	yes	—
166	D₄	—	2	4	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	2	no	—
169	D₄	C₂²×Dic₃₁	1	4	yes	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	62	2	no	—
174	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
175	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
176	C₂×C₄	#92497	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
180	C₈	—	31	4	no	—
181	C₈	—	62	4	no	—
182	C₈	C₂²×Dic₃₁	1	1	yes	characteristic
183	C₈	C₂²×Dic₃₁	1	1	yes	characteristic
184	C₈	C₂²×Dic₃₁	1	1	yes	characteristic
185	C₈	C₂²×Dic₃₁	1	1	yes	characteristic
186	C₂²	—	2	4	no	—
187	C₂²	—	4	1	no	—
188	C₂²	—	8	2	no	—
189	C₂²	D₄×Dic₃₁	1	1	yes	characteristic
190	C₂²	D₄×Dic₃₁	1	1	yes	characteristic
191	C₂²	C₂×#92497	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	124	2	no	—
194	C₄	D₄×Dic₃₁	1	1	yes	characteristic
195	C₄	D₄×Dic₃₁	1	1	yes	characteristic
196	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
197	C₄	C₂³×Dic₃₁	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	4	no	—
202	C₂	C₂×D₄×Dic₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₃₁
commutator	C₄×C₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₁
Fitting, prime radical	C₂×C₃₁×#135	C₂
socle	C₂×C₃₁	C₂×C₄×D₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	C₂×#135	D₃₁
hypocenter	C₃₁	#2957
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #14521816	C₂	semidirect
2. #14521171	C₂	semidirect
3. #14521166	C₂	semidirect
4. C₂×#11758407	C₂	semidirect
5. #14521066	C₂	semidirect
6. #14521004	C₂	semidirect
9. C₃₁×#135	C₄	semidirect
10. C₃₁×#135	C₄	semidirect
13. C₂×Q₁₆×C₃₁	C₄	semidirect
14. C₂×SD₁₆×C₃₁	C₄	semidirect
15. C₂×D₈×C₃₁	C₄	semidirect
18. #11758525	C₂²	semidirect
23. #11758425	C₂²	semidirect
24. #11758407	C₂²	semidirect
25. #11758407	C₂²	semidirect
26. #11758403	C₂²	semidirect
27. C₈×Dic₃₁	C₂²	semidirect
110. #135	Dic₃₁	semidirect
111. #135	Dic₃₁	semidirect
114. C₂×Q₁₆	Dic₃₁	semidirect
115. C₂×SD₁₆	Dic₃₁	semidirect
116. C₂×D₈	Dic₃₁	semidirect
129. C₃₁	#2957	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14521816	2
#14521171	2
#14521166	2
C₂×#11758407	2
#14521066	2
#14521004	2
C₂×C₃₁×#135	2
#2957	31